| L(s) = 1 | + 5-s + 7-s − 3·9-s − 2·13-s − 2·17-s + 4·19-s − 23-s + 25-s + 2·29-s + 35-s − 2·37-s + 2·41-s + 4·43-s − 3·45-s − 12·47-s + 49-s − 2·53-s − 12·59-s + 6·61-s − 3·63-s − 2·65-s − 4·67-s − 10·73-s + 4·79-s + 9·81-s − 12·83-s − 2·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 9-s − 0.554·13-s − 0.485·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s + 0.371·29-s + 0.169·35-s − 0.328·37-s + 0.312·41-s + 0.609·43-s − 0.447·45-s − 1.75·47-s + 1/7·49-s − 0.274·53-s − 1.56·59-s + 0.768·61-s − 0.377·63-s − 0.248·65-s − 0.488·67-s − 1.17·73-s + 0.450·79-s + 81-s − 1.31·83-s − 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.762831845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.762831845\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.46481091023042, −13.96577714793542, −13.65389193340587, −12.94065646786781, −12.45868865420194, −11.82864203561103, −11.43274821361845, −10.98765692963618, −10.32120412370617, −9.812646217247194, −9.297529213552846, −8.718280114590484, −8.306890890728082, −7.566451171013696, −7.222372807432290, −6.276650109625195, −6.063104892031918, −5.224335783900416, −4.915363249423399, −4.211208613572269, −3.286443427203472, −2.858945635314449, −2.128163545459949, −1.451493079845363, −0.4571574666915683,
0.4571574666915683, 1.451493079845363, 2.128163545459949, 2.858945635314449, 3.286443427203472, 4.211208613572269, 4.915363249423399, 5.224335783900416, 6.063104892031918, 6.276650109625195, 7.222372807432290, 7.566451171013696, 8.306890890728082, 8.718280114590484, 9.297529213552846, 9.812646217247194, 10.32120412370617, 10.98765692963618, 11.43274821361845, 11.82864203561103, 12.45868865420194, 12.94065646786781, 13.65389193340587, 13.96577714793542, 14.46481091023042
